Let X and Y be discrete random variables that take the values 1,2, ... N with equal probability, where N is a positive integer. X and Y are independent, so in particular P(X ≤a and Y ≤ b) = P(X ≤ a)P(Y≤ b). Let Z max (X, Y). Note that Z is also a random variable that takes = values from 1 to N. a. Write down the probability distribution function for X (which is also the probability distribution function for Y). b. Find the expected value of X c. Find P(Z < 20) (Hint: Your answer will involve the parameter N. There are two cases: what are they? Note also that max (X, Y) ≤ 20 if and only if both X ≤ 20 and Y ≤ 20). d. Find the cdf of Z. (Hint: Write down the definition of F₂(x).) e. Using the cdf from part d., find the probability distribution function for Z. (NOTE I am being very specific here. You MUST use the cdf of to do this problem. If you do it a different way, you receive no credit.) f. Find EIZ). (Hint: you will need the formulas for E-₁n and -₁²) n n=1 E[Z] g. Find limy co 100 E[X]' and comment on your answer. Can you think of a practical situation in which this information might be useful?